Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

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June 2016, 11(2): 203-222. doi: 10.3934/nhm.2016.11.203

Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws

Boris Andreianov ^1, and Mohamed Karimou Gazibo ^2,

1.	Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 25030 Besançon Cedex
2.	Université Abdou Moumouni de Niamey, École Normale Supérieure, Departement de Mathématiques, BP: 10963 Niamey, Niger

Received May 2015 Revised October 2015 Published March 2016

Abstract
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We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.

Keywords: weakly trace-regular solutions, Degenerate parabolic-hyperbolic equation, strong entropy formulation, uniqueness proof., Dirichlet boundary condition.

Mathematics Subject Classification: Primary: 35M13; Secondary: 35L0.

Citation: Boris Andreianov, Mohamed Karimou Gazibo. Explicit formulation for the Dirichlet problem for parabolic-hyperbolic conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 203-222. doi: 10.3934/nhm.2016.11.203

References:

[1]	J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations,, J. Hyperb. Differ. Equ., 10 (2013), 659. doi: 10.1142/S0219891613500239.
[2]	K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function,, J. Differ. Equ., 228* (2006), 111. doi: 10.1016/j.jde.2006.05.002.*
[3]	B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperb. Differ. Equ., 7 (2010), 1. doi: 10.1142/S0219891610002062.
[4]	B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition,, J. Evol. Equ., 4 (2004), 273. doi: 10.1007/s00028-004-0143-1.
[5]	B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition,, Z. Angew. Math. Phys., 64 (2013), 1471. doi: 10.1007/s00033-012-0297-6.
[6]	B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition，, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, 77 (2014), 303. doi: 10.1007/978-3-319-05684-5_29.
[7]	B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions,, C. R. Acad. Paris Ser. I Math., 345* (2007), 431. doi: 10.1016/j.crma.2007.09.008.*
[8]	B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws,, Trans. AMS, 367* (2015), 3763. doi: 10.1090/S0002-9947-2015-05988-1.*
[9]	C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. PDE, 4 (1979), 1017. doi: 10.1080/03605307908820117.
[10]	Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications,, Thèse d'état, (1972).
[11]	Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().
[12]	Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313.
[13]	R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition,, J. Math. Anal. Appl., 326* (2007), 108. doi: 10.1016/j.jmaa.2006.02.072.*
[14]	R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration,, SIAM J. Math. Anal., 34 (2003), 611.
[15]	C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media,, Interfaces Free Bound, 11 (2009), 239. doi: 10.4171/IFB/210.
[16]	J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Ration. Mech. Anal., 147* (1999), 269. doi: 10.1007/s002050050152.*
[17]	G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147* (1999), 89. doi: 10.1007/s002050050146.*
[18]	R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains,, Acta Mathematica Sci., 35 (2015), 906. doi: 10.1016/S0252-9602(15)30028-X.
[19]	F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws,, J. Differ. Equ., 71 (1988), 93. doi: 10.1016/0022-0396(88)90040-X.
[20]	S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1838. doi: 10.1137/S0036142998336138.
[21]	G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere,, Math. et Appl., 22 (1996).
[22]	M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition,, In F. Ancona et al., 8 (2014), 583.
[23]	M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().
[24]	M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites,, Thèse de Doctorat Besançon, (2013).
[25]	S. N. Kruzhkov, First order quasi-linear equations in several independent variables,, Math. USSR Sb., 10 (1970), 217.
[26]	Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations,, Discrete Contin. Dyn. Syst., 25 (2009), 1275. doi: 10.3934/dcds.2009.25.1275.
[27]	M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems,, J. Evol. Equ., 3 (2003), 603. doi: 10.1007/s00028-003-0105-z.
[28]	F. Otto, Initial-boundary value problem for a scalar conservation laws,, C. R. Acad. Sci. Paris Sér I Math., 322* (1996), 729.*
[29]	C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Ration. Mech. Anal., 163* (2002), 87. doi: 10.1007/s002050200184.*
[30]	A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262. doi: 10.1137/S0036142902406612.
[31]	E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions,, Iszvestiya Math., 66 (2002), 1171. doi: 10.1070/IM2002v066n06ABEH000411.
[32]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws,, J. Hyp. Diff. Equ., 4 (2009), 729. doi: 10.1142/S0219891607001343.
[33]	E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux,, J. Differ. Equ., 247* (2009), 2821. doi: 10.1016/j.jde.2009.08.022.*
[34]	A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains,, Comm. PDEs, 28 (2003), 381. doi: 10.1081/PDE-120019387.
[35]	E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées,, An. Fac. Sci. Toulouse, 10 (2001), 163. doi: 10.5802/afst.987.
[36]	T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation,, Math. Res. Lett., 3 (1996), 475. doi: 10.4310/MRL.1996.v3.n4.a6.
[37]	G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation,, Advance in Math. Sci. Appl., 15 (2005), 423.
[38]	A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160* (2001), 181. doi: 10.1007/s002050100157.*
[39]	A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations,, Math. USSR Sbornik, 78 (1969), 374.
[40]	J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563. doi: 10.1007/s002110100307.

show all references

References:

[1]	J. Aleksic and D. Mitrovic, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations,, J. Hyperb. Differ. Equ., 10 (2013), 659. doi: 10.1142/S0219891613500239.
[2]	K. Ammar, P. Wittbold and J. Carrillo, Scalar conservation laws with general boundary condition and continuous flux function,, J. Differ. Equ., 228* (2006), 111. doi: 10.1016/j.jde.2006.05.002.*
[3]	B. Andreianov, M. Bendahmane and K. H. Karlsen, Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations,, J. Hyperb. Differ. Equ., 7 (2010), 1. doi: 10.1142/S0219891610002062.
[4]	B. Andreianov and F. Bouhsiss, Uniqueness for an elliptic-parabolic problem with Neumann boundary condition,, J. Evol. Equ., 4 (2004), 273. doi: 10.1007/s00028-004-0143-1.
[5]	B. Andreianov and M. Karimou Gazibo, Entropy formulation of degenerate parabolic equation with zero-flux boundary condition,, Z. Angew. Math. Phys., 64 (2013), 1471. doi: 10.1007/s00033-012-0297-6.
[6]	B. Andreianov and M. Karimou Gazibo, Convergence of finite volume scheme for degenerate parabolic problem with zero flux boundary condition，, Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, 77 (2014), 303. doi: 10.1007/978-3-319-05684-5_29.
[7]	B. Andreianov and K. Shibi, Scalar conservation laws with nonlinear boundary conditions,, C. R. Acad. Paris Ser. I Math., 345* (2007), 431. doi: 10.1016/j.crma.2007.09.008.*
[8]	B. Andreianov and K. Sbihi, Well-posedness of general boundary-value problems for scalar conservation laws,, Trans. AMS, 367* (2015), 3763. doi: 10.1090/S0002-9947-2015-05988-1.*
[9]	C. Bardos, A.-Y. LeRoux and J.-C. Nédélec, First order quasilinear equations with boundary conditions,, Comm. PDE, 4 (1979), 1017. doi: 10.1080/03605307908820117.
[10]	Ph. Bénilan, Equations D'évolution dans un Espace de Banach Quelconque et Applications,, Thèse d'état, (1972).
[11]	Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces,, Preprint book., ().
[12]	Ph. Bénilan, J. Carrillo and P. Wittbold, Renormalized entropy solutions of scalar conservation laws,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 313.
[13]	R. Bürger, H. Frid and K.H. Karlsen, On the well-posedness of entropy solution to conservation laws with a zero-flux boundary condition,, J. Math. Anal. Appl., 326* (2007), 108. doi: 10.1016/j.jmaa.2006.02.072.*
[14]	R. Bürger, H. Frid and K. H. Karlsen, On a free boundary problem for a strongly degenerate quasilinear parabolic equation with an application to a model of presssure filtration,, SIAM J. Math. Anal., 34 (2003), 611.
[15]	C. Cancès, Th. Gallouët and A. Porretta, Two-phase flows involving capillary barriers in heterogeneous porous media,, Interfaces Free Bound, 11 (2009), 239. doi: 10.4171/IFB/210.
[16]	J. Carrillo, Entropy solutions for nonlinear degenerate problems,, Arch. Ration. Mech. Anal., 147* (1999), 269. doi: 10.1007/s002050050152.*
[17]	G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147* (1999), 89. doi: 10.1007/s002050050146.*
[18]	R. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains,, Acta Mathematica Sci., 35 (2015), 906. doi: 10.1016/S0252-9602(15)30028-X.
[19]	F. Dubois and Ph. LeFloch, Boundary condition for nonlinear hyperbolic conservation laws,, J. Differ. Equ., 71 (1988), 93. doi: 10.1016/0022-0396(88)90040-X.
[20]	S. Evje and K. H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations,, SIAM J. Numer. Anal., 37 (2000), 1838. doi: 10.1137/S0036142998336138.
[21]	G. Gagneux and M. Madaune-Tort, Analyse Mathématique de Modeles Nonlinéaires de L'ingénierie Pétroliere,, Math. et Appl., 22 (1996).
[22]	M. Karimou Gazibo, Degenerate Convection-Diffusion Equation with a Robin boundary condition,, In F. Ancona et al., 8 (2014), 583.
[23]	M. Karimou Gazibo, Degenerate parabolic equation with zero flux boundary condition and its approximations,, Preprint available at , ().
[24]	M. Karimou Gazibo, Études Mathématiques et Numériques des Problèmes Paraboliques Avec Des Conditions Aux Limites,, Thèse de Doctorat Besançon, (2013).
[25]	S. N. Kruzhkov, First order quasi-linear equations in several independent variables,, Math. USSR Sb., 10 (1970), 217.
[26]	Y. S. Kwon, Strong traces for degenerate parabolic-hyperbolic equations,, Discrete Contin. Dyn. Syst., 25 (2009), 1275. doi: 10.3934/dcds.2009.25.1275.
[27]	M. Maliki and H. Touré, Uniqueness of entropy solutions for nonlinear degenerate parabolic problems,, J. Evol. Equ., 3 (2003), 603. doi: 10.1007/s00028-003-0105-z.
[28]	F. Otto, Initial-boundary value problem for a scalar conservation laws,, C. R. Acad. Sci. Paris Sér I Math., 322* (1996), 729.*
[29]	C. Mascia, A. Porretta and A. Terracina, Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equation,, Arch. Ration. Mech. Anal., 163* (2002), 87. doi: 10.1007/s002050200184.*
[30]	A. Michel and J. Vovelle, Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods,, SIAM J. Numer. Anal., 41 (2003), 2262. doi: 10.1137/S0036142902406612.
[31]	E. Yu. Panov, On the theory of generalized entropy solutions of Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions,, Iszvestiya Math., 66 (2002), 1171. doi: 10.1070/IM2002v066n06ABEH000411.
[32]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multi-dimensional scalar conservation laws,, J. Hyp. Diff. Equ., 4 (2009), 729. doi: 10.1142/S0219891607001343.
[33]	E. Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux,, J. Differ. Equ., 247* (2009), 2821. doi: 10.1016/j.jde.2009.08.022.*
[34]	A. Porretta and J. Vovelle, $L^1$ solutions to first order hyperbolic equations in bounded domains,, Comm. PDEs, 28 (2003), 381. doi: 10.1081/PDE-120019387.
[35]	E. Rouvre and G. Gagneux, Formulation forte entropique de lois scalaires hyperboliques-paraboliques dégénérées,, An. Fac. Sci. Toulouse, 10 (2001), 163. doi: 10.5802/afst.987.
[36]	T. Tassa, Regularity of weak solutions of the nonlinear Fokker-Planck equation,, Math. Res. Lett., 3 (1996), 475. doi: 10.4310/MRL.1996.v3.n4.a6.
[37]	G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation,, Advance in Math. Sci. Appl., 15 (2005), 423.
[38]	A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160* (2001), 181. doi: 10.1007/s002050100157.*
[39]	A. I. Vol'pert and S. I. Hudjaev, Cauchy problem for degenerate second order quasilinear parabolic equations,, Math. USSR Sbornik, 78 (1969), 374.
[40]	J. Vovelle, Convergence of finite volume monotones schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563. doi: 10.1007/s002110100307.

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